Let l ,   m ,   n ∈ R and A = 1 r r 2 l r r 2 1 m r 2 1 r n . Then the set of all real values of r for which the rank of A is 3 , is

Step-by-step solution

A = 1 r r 2 l r r 2 1 m r 2 1 r n Let l ,   m ,   n ∈ R ⇒ l ,   m ,   n = 1 ⇒ A =   1 r r 2 1 r r 2 1 1 r 2 1 r 1 Checking options 1. 0 , ∞ Here, r = any value which lies between ( 0 , ∞ ) . Case ( i ) : Let r = 1 A = 1 1 1 1 1 1 1 1 1 1 1 1 R 2 → R 2 - R 1 ,   R 3 → R 3 - R 1 = 1 1 1 1 0 0 0 0 0 0 0 0 ∴ Rank of A is 1 ∴   R ∉ 1 Case ( ii ) : Let r = 2 A = 1 2 4 1 2 4 1 1 4 1 2 1 If r = 2 , rank of A is 3 .